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Finding the Least Common Denominator

What is the Least Common Denominator

A denominator is the bottom number of a fraction, or the number below the fraction line. For a fraction 1/3, the denominator is 3.

A common denominator is a denominator that is common to the fractions being operated on. For the denominator to be common it must be the same in all fractions. For instance, 1/3 and 2/3 have common denominators, and 1/3 and 2/5 do not have common denominators.

The least common denominator is the smallest common denominator. It is the smallest number that is evenly divisible by all uncommon denominators. The least common denominator for the fractions 1/3 and 2/5 is 15. 15 ÷ 3 = 5 and 15 ÷ 3 = 5. Note that there can be no remainder when dividing by one of the denominators by the least common denominator. The least common denominator is also referred to as the lowest common denominator or the least common multiple.

How to Find the Least Common Denominator Using Factorization

One way to find the least common denominator is to use prime factorization. Find the prime factors of each denominator. Then multiply all of the prime factors, multiplying the factors that are common to both only once, to find the least common denominator. Also see our factors calculator to find the factors of your denominators, including the greatest common factor.

Example 1: Finding the Least Common Denominator of 36 and 90

Find the prime factors of 36

Divide 36 by 2, which equals 18. 18 and 2 are factors. 18 can be factored again into 6 and 3. 6 can be factored into 2 and 3. The prime factors are thus [3, 3, 2, 2].

Find the prime factors of 90

Divide 90 by 10, which equals 9. 9 and 10 are factors. 9 can be factored again into 3 and 3. 10 can be factored into 5 and 2. The prime factors are thus [5, 3, 3, 2].

The prime factors of 36 and 90 are [5, 3, 3, 2, 2]. Note that 3, 3, and 2 are common between 36 and 90 so they are only used once.

The least common denominator = 5 × 3 × 3 × 3 × 2 × 2 = 180.

Example 2: Finding the Least Common Denominator of 105 and 165

Find the prime factors of 105

Divide 105 by 7, which equals 15. 15 and 7 are factors. 15 can be factored again into 5 and 3. The prime factors are thus [7, 5, 3].

Find the prime factors of 165

Divide 165 by 11 which equals 15. 15 and 11 are factors. 15 can be factored again into 5 and 3. The prime factors are thus [11, 5, 3].

The prime factors of 105 and 165 are [11, 7, 5, 3]. Note that 5 and 3 are common between 105 and 165 so they are only used once.

The least common denominator = 11 × 7 × 5 × 3 = 1155.

Example 3: Finding the Least Common Denominator of 24 and 42

Find the prime factors of 24

Divide 24 by 2, which equals 12. 12 and 2 are factors. 12 can be factored again into 6 and 2. 6 can be factored again into 2 and 3. The prime factors are thus [3, 2, 2, 2].

Find the prime factors of 42

Divide 42 by 7 which equals 6. 7 and 6 are factors. 6 can be factored again into 2 and 3. The prime factors are thus [7, 3, 2].

The prime factors for 24 and 42 are [7, 3, 2, 2, 2]. Note that 3 and 2 are common between 24 and 42 so they are only used once.

The least common denominator = 7 × 3 × 2 × 2 × 2 = 168.

Finding the Least Common Denominator by Finding all Multiples

You can also find the least common denominator by finding all of the multiples of each denominator and finding the smallest multiple that is common to both. Continuing the example above:

The multiples of 3 are [3,6,9,12,15,18,21,24,27,39,…]

The multiples of 5 are [5,10,15,20,25,30,…]

Note that the multiples that are common to both 3 and 5 are 15 and 30. The smallest common multiple is 15, which makes it the least common denominator.